What does a z-score actually tell me?

How many standard deviations a value sits from the mean, which makes values from different distributions comparable. A z of 2 means the value is in the top 2.3% if the distribution is normal.

Does a z-score require a normal distribution?

The score itself does not; it is just a rescaling and can be computed for any distribution. Reading a probability off it does, because the percentile lookup assumes normality. Applying it to a skewed distribution is where it goes wrong.

Math & Statisticsstatistics

Z-Score Calculator

Z-score with the percentile and two-tailed p-value it implies, using the normal CDF.

Also called: standard score calculator, how many standard deviations.

Z-score

1.5

A z-score of 1.5. 85 sits 1.5 standard deviations from the mean, at the 93.32%th percentile.

Percentile

93.32%

Two-tailed p-value

13.36%

How this is calculated

Method and background

How this is calculated

The distance from the mean measured in standard deviations. Converting that to a percentile assumes the underlying distribution is normal, which is an assumption, not a fact, and the reason a z-score on visibly skewed data is misleading.

z = (value - mean) / sd

value: The value
mean: Population mean
sd: Population standard deviation

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

1.5 standard deviations above

Value: 85
Mean: 70
Standard deviation: 10

Z-score1.5

standard normal table: Phi(1.5) = 0.93319

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the mean itself is z = 0

Value: 70
Mean: 70
Standard deviation: 10

Z-score0

boundary

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zero standard deviation is undefined

Value: 85
Mean: 70
Standard deviation: 0

Z-score0

degenerate case: guarded division

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Frequently asked questions

What does a z-score actually tell me?: How many standard deviations a value sits from the mean, which makes values from different distributions comparable. A z of 2 means the value is in the top 2.3% if the distribution is normal.
Does a z-score require a normal distribution?: The score itself does not; it is just a rescaling and can be computed for any distribution. Reading a probability off it does, because the percentile lookup assumes normality. Applying it to a skewed distribution is where it goes wrong.